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Hi, I'm Fiona,
welcome to TMUA 2016 paper two,

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question 3 and the question asks us what
is the value in radians of the largest

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angle X in the range 0 to 2π that
satisfies the equation 8 sin squared X +

ac408031-1aaf-4e52-a841-81624ba3648c-3
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4 Cos squared X = 7.

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So let's start with the equation itself.

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In its current form I can't solve it,
but I'm going to use this identity sin

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squared X plus Cos squared X = 1 to
rearrange it to get Cos squared X in

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terms of sin squared X,
and then substitute that into the

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equation and I end up with 8 sin squared
X + 4 - 4 sin greater equals 7.

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Collecting like terms together,
I now arrive in this form 4 sin squared X

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- 3 = 0 and it may not be obvious at
first,

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but this is in fact a difference of two
squares.

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I've got the difference because I've got
my - there.

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Sin squared X is already in its squared
form, 4 is 2 squared,

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and of course 3 is also the same value as
root 3 squared.

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So now I'm going to use my difference of
two squares to factorise and I end up

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with this factorisation and this will
give me two equations in sine X.

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So I have sin X equals root 3/2,
or sin X equals minus root 3/2.

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At this point you might have a sketch of
the graph of sin X in your mind already,

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but it is a good use of time to get the
graph even if you're under exam

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conditions and time pressure,
because then we can see much more clearly

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and we can be confident.

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So looking back at the range of values of
X that I'm interested in,

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I only really need to consider this side
of my graph of sin X.

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And for me I like to think of this
triangle.

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So I need to find out what value of X
satisfies sin X equals root 2/2.

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Now you might remember this from GCSE
maths,

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but if you don't you can remember a
triangle like this which will help you

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because we know that sin of an angle
equals the opposite over the hypotenuse.

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So in this case the opposite being root 3
and the hypotenuse being 2.

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I can be confident that my angle X is
going to be π/3.

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And if I'm thinking about 1/3 of π,
then I'm looking at my sketch of the

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graph of sin X and I can see that I
should expect that to be a solution

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around about here,
another one around about here.

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And then I'm also considering that sin X
also can equal minus root 3/2.

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So I would expect to see another solution
around about here,

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and a final solution within this range
around about here.

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Now coming back to this triangle,
if you find it a bit more tricky to

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memorise things,
all you need to know is that this

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triangle originates from an equilateral
triangle of side length 2.

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So that might help you remember it.

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But back to the question itself.

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Using the information that I have,
I know now that my possible values of X

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that satisfy this equation are are π/3,
2π/3, 4π/3, and 5π/3.

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And of course the question asks us for
the largest of these,

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so that's going to be 5π/3.

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And my correct answer is D.